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{"name": "Differentiation of Basic Polynomial Expressions (with second derivatives)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"extensions": [], "statement": "<p><strong>Differentiate the following polynomials.</strong></p>\n<p><em>Note:&nbsp;</em>some questions may not&nbsp;include all the possible terms.</p>", "ungrouped_variables": ["ac", "bc", "cc", "dc", "ec", "fc", "ap", "bp", "cp", "dp"], "parts": [{"gaps": [{"showFeedbackIcon": true, "correctAnswerStyle": "plain", "correctAnswerFraction": false, "allowFractions": false, "variableReplacementStrategy": "originalfirst", "notationStyles": ["plain", "en", "si-en"], "marks": "0.3", "minValue": "{ac[0]}*{ap}", "variableReplacements": [], "type": "numberentry", "scripts": {}, "maxValue": "{ac[0]}*{ap}", "showCorrectAnswer": true}, {"showFeedbackIcon": true, "correctAnswerStyle": "plain", "correctAnswerFraction": false, "allowFractions": false, "variableReplacementStrategy": "originalfirst", "notationStyles": ["plain", "en", "si-en"], "marks": "0.3", "minValue": "{bc[0]}*{bp}", "variableReplacements": [], "type": "numberentry", "scripts": {}, "maxValue": "{bc[0]}*{bp}", "showCorrectAnswer": true}, {"showFeedbackIcon": true, "correctAnswerStyle": "plain", "correctAnswerFraction": false, "allowFractions": false, "variableReplacementStrategy": "originalfirst", "notationStyles": ["plain", "en", "si-en"], "marks": "0.2", "minValue": "{cc[0]}*{cp}", "variableReplacements": [], "type": "numberentry", "scripts": {}, "maxValue": "{cc[0]}*{cp}", "showCorrectAnswer": true}, {"showFeedbackIcon": true, "correctAnswerStyle": "plain", "correctAnswerFraction": false, "allowFractions": false, "variableReplacementStrategy": "originalfirst", "notationStyles": ["plain", "en", "si-en"], "marks": "0.2", "minValue": "{dc[0]}*{dp}", "variableReplacements": [], "type": "numberentry", "scripts": {}, "maxValue": "{dc[0]}*{dp}", "showCorrectAnswer": true}, {"showFeedbackIcon": true, "correctAnswerStyle": "plain", "correctAnswerFraction": false, "allowFractions": false, "variableReplacementStrategy": "originalfirst", "notationStyles": ["plain", "en", "si-en"], "marks": "0.2", "minValue": "{ec[0]}", "variableReplacements": [], "type": "numberentry", "scripts": {}, "maxValue": "{ec[0]}", "showCorrectAnswer": true}, {"showFeedbackIcon": true, "integerPartialCredit": 0, "correctAnswerFraction": false, "allowFractions": false, "variableReplacementStrategy": "originalfirst", "correctAnswerStyle": "plain", "notationStyles": ["plain", "en", "si-en"], "marks": "0.2", "minValue": "{ac[0]}*5*4", "variableReplacements": [], "type": "numberentry", "integerAnswer": true, "scripts": {}, "maxValue": "{ac[0]}*5*4", "showCorrectAnswer": true}, {"showFeedbackIcon": true, "integerPartialCredit": 0, "correctAnswerFraction": false, "allowFractions": false, "variableReplacementStrategy": "originalfirst", "correctAnswerStyle": "plain", "notationStyles": ["plain", "en", "si-en"], "marks": "0.2", "minValue": "{bc[0]}*4*3", "variableReplacements": [], "type": "numberentry", "integerAnswer": true, "scripts": {}, "maxValue": "{bc[0]}*4*3", "showCorrectAnswer": true}, {"showFeedbackIcon": true, "integerPartialCredit": 0, "correctAnswerFraction": false, "allowFractions": false, "variableReplacementStrategy": "originalfirst", "correctAnswerStyle": "plain", "notationStyles": ["plain", "en", "si-en"], "marks": "0.2", "minValue": "{cc[0]}*3*2", "variableReplacements": [], "type": "numberentry", "integerAnswer": true, "scripts": {}, "maxValue": "{cc[0]}*3*2", "showCorrectAnswer": true}, {"showFeedbackIcon": true, "integerPartialCredit": 0, "correctAnswerFraction": false, "allowFractions": false, "variableReplacementStrategy": "originalfirst", "correctAnswerStyle": "plain", "notationStyles": ["plain", "en", "si-en"], "marks": "0.2", "minValue": "{dc[0]}*2", "variableReplacements": [], "type": "numberentry", "integerAnswer": true, "scripts": {}, "maxValue": "{dc[0]}*2", "showCorrectAnswer": true}], "showCorrectAnswer": true, "marks": 0, "variableReplacements": [], "type": "gapfill", "prompt": "<p>$\\simplify[basic,zeroterm,zerofactor,unitfactor]{{ac[0]}x^5+{bc[0]}x^4+{cc[0]}x^3+{dc[0]}x^2+{ec[0]}x+{fc[0]}}$</p>\n<p>$f'(x)=$[[0]]$x^4+$[[1]]$x^3+$[[2]]$x^2+$[[3]]$x+$[[4]]</p>\n<p>$f''(x)=$[[5]]$x^3+$[[6]]$x^2+$[[7]]$x+$[[8]]</p>", "scripts": {}, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true}, {"gaps": [{"showFeedbackIcon": true, "showpreview": true, "vsetrangepoints": 5, "expectedvariablenames": [], "variableReplacementStrategy": "originalfirst", "vsetrange": [0, 1], "answersimplification": "all", "marks": "1", "checkingtype": "absdiff", "checkingaccuracy": 0.001, "variableReplacements": [], "answer": "{ac[1]}*{ap}*x^{ap-1}+{bc[1]}*{bp}*x^{bp-1}+{cc[1]}*{cp}*x^{cp-1}+{dc[1]}*{dp}*x^{dp-1}+{ec[1]}", "type": "jme", "scripts": {}, "checkvariablenames": false, "showCorrectAnswer": true}, {"showFeedbackIcon": true, "showpreview": true, "vsetrangepoints": 5, "expectedvariablenames": [], "variableReplacementStrategy": "originalfirst", "vsetrange": [0, 1], "answersimplification": "all", "marks": "1", "checkingtype": "absdiff", "checkingaccuracy": 0.001, "variableReplacements": [], "answer": "{ac[1]}*{ap}*4*x^{ap-2}+{bc[1]}*{bp}*3*x^{bp-2}+{cc[1]}*{cp}*2*x^{cp-2}+{dc[1]}*{dp}", "type": "jme", "scripts": {}, "checkvariablenames": false, "showCorrectAnswer": true}], "showCorrectAnswer": true, "marks": 0, "variableReplacements": [], "type": "gapfill", "prompt": "<p>$\\simplify[basic,zeroterm,zerofactor,unitfactor]{{ac[1]}x^{ap}+{bc[1]}x^{bp}+{cc[1]}x^{cp}+{dc[1]}x^{dp}+{ec[1]}x+{fc[1]}}$</p>\n<p>$f'(x)=$&nbsp;[[0]]</p>\n<p>$f''(x)=$&nbsp;[[1]]</p>", "scripts": {}, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true}, {"gaps": [{"showFeedbackIcon": true, "showpreview": true, "vsetrangepoints": 5, "expectedvariablenames": [], "variableReplacementStrategy": "originalfirst", "vsetrange": [0, 1], "answersimplification": "all", "marks": "1", "checkingtype": "absdiff", "checkingaccuracy": 0.001, "variableReplacements": [], "answer": "{ac[2]}*{ap}*x^{ap-1}+{bc[2]}*{bp}*x^{bp-1}+{cc[2]}*{cp}*x^{cp-1}+{dc[2]}*{dp}*x^{dp-1}+{ec[2]}", "type": "jme", "scripts": {}, "checkvariablenames": false, "showCorrectAnswer": true}, {"showFeedbackIcon": true, "showpreview": true, "vsetrangepoints": 5, "expectedvariablenames": [], "variableReplacementStrategy": "originalfirst", "vsetrange": [0, 1], "answersimplification": "all", "marks": "1", "checkingtype": "absdiff", "checkingaccuracy": 0.001, "variableReplacements": [], "answer": "{ac[2]}*{ap}*4*x^{ap-2}+{bc[2]}*{bp}*3*x^{bp-2}+{cc[2]}*{cp}*2*x^{cp-2}+{dc[2]}*{dp}", "type": "jme", "scripts": {}, "checkvariablenames": false, "showCorrectAnswer": true}], "showCorrectAnswer": true, "marks": 0, "variableReplacements": [], "type": "gapfill", "prompt": "<p>$\\simplify[basic,zeroterm,zerofactor,unitfactor]{{ac[2]}x^{ap}+{bc[2]}x^{bp}+{cc[2]}x^{cp}+{dc[2]}x^{dp}+{ec[2]}x+{fc[2]}}$</p>\n<p>$f'(x)=$&nbsp;[[0]]</p>\n<p>$f''(x)=$&nbsp;[[1]]</p>", "scripts": {}, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true}, {"gaps": [{"showFeedbackIcon": true, "showpreview": true, "vsetrangepoints": 5, "expectedvariablenames": [], "variableReplacementStrategy": "originalfirst", "vsetrange": [0, 1], "answersimplification": "all", "marks": "1", "checkingtype": "absdiff", "checkingaccuracy": 0.001, "variableReplacements": [], "answer": "{ac[3]}*{ap}*x^{ap-1}+{bc[3]}*{bp}*x^{bp-1}+{cc[3]}*{cp}*x^{cp-1}+{dc[3]}*{dp}*x^{dp-1}+{ec[3]}", "type": "jme", "scripts": {}, "checkvariablenames": false, "showCorrectAnswer": true}, {"showFeedbackIcon": true, "showpreview": true, "vsetrangepoints": 5, "expectedvariablenames": [], "variableReplacementStrategy": "originalfirst", "vsetrange": [0, 1], "answersimplification": "all", "marks": "1", "checkingtype": "absdiff", "checkingaccuracy": 0.001, "variableReplacements": [], "answer": "{ac[3]}*{ap}*4*x^{ap-2}+{bc[3]}*{bp}*3*x^{bp-2}+{cc[3]}*{cp}*2*x^{cp-2}+{dc[3]}*{dp}", "type": "jme", "scripts": {}, "checkvariablenames": false, "showCorrectAnswer": true}], "showCorrectAnswer": true, "marks": 0, "variableReplacements": [], "type": "gapfill", "prompt": "<p>$\\simplify[basic,zeroterm,zerofactor,unitfactor]{{ac[3]}x^{ap}+{bc[3]}x^{bp}+{cc[3]}x^{cp}+{dc[3]}x^{dp}+{ec[3]}x+{fc[3]}}$</p>\n<p>$f'(x)=$&nbsp;[[0]]</p>\n<p>$f''(x)=$&nbsp;[[1]]</p>", "scripts": {}, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true}], "tags": [], "rulesets": {}, "advice": "<p style=\"text-align: center;\">If $y=ax^n$,</p>\n<p style=\"text-align: center;\">$\\frac{dy}{dx}=anx^{n-1}$ for all rational $n$.</p>\n<p style=\"text-align: left;\">We'll take one of the terms from Part a as an example:</p>\n<p style=\"text-align: left;\">$\\var{cc[0]}x^\\var{cp}$</p>\n<p style=\"text-align: left;\">All we have to do to terms where $x$ is to a power of anything is times the coefficient of $x$ by the original power, and then take one away from the original power.</p>\n<p style=\"text-align: left;\">If you are not familiar with this kind of work, these instructions may sound confusing, but it is much easier once you have seen it in practice.</p>\n<p style=\"text-align: left;\">We take</p>\n<p style=\"text-align: left;\">$\\var{cc[0]}x^\\var{cp}$</p>\n<p style=\"text-align: left;\">and times $\\var{cc[0]}$ by $\\var{cp}$, to get</p>\n<p style=\"text-align: left;\">$(\\var{cc[0]}*\\var{cp})x^\\var{cp}=\\simplify{{cc[0]}*{cp}x^{cp}}$.</p>\n<p style=\"text-align: left;\">We then subtract one from the original power, $\\var{cp}$.</p>\n<p style=\"text-align: left;\">This gives us the final answer of</p>\n<p style=\"text-align: left;\"><strong>$\\simplify{{cc[0]}*{cp}x^{cp-1}}$</strong>.</p>\n<p style=\"text-align: left;\"></p>\n<p style=\"text-align: left;\">Remember, don't be confused if there is no coefficient. The fact the term is there means the coefficient must be $1$, but we don't tend to write it out as, for example $1x$, we just say $x$.</p>", "variables": {"dp": {"templateType": "anything", "group": "Ungrouped variables", "name": "dp", "definition": "2", "description": ""}, "bc": {"templateType": "anything", "group": "Ungrouped variables", "name": "bc", "definition": "repeat(random(-6..6),4)", "description": ""}, "ac": {"templateType": "anything", "group": "Ungrouped variables", "name": "ac", "definition": "repeat(random(-3..3),4)", "description": ""}, "cc": {"templateType": "anything", "group": "Ungrouped variables", "name": "cc", "definition": "repeat(random(-10..10 except 0 except 1 except -1),4)", "description": ""}, "ec": {"templateType": "anything", "group": "Ungrouped variables", "name": "ec", "definition": "repeat(random(-30..30),4)", "description": ""}, "bp": {"templateType": "anything", "group": "Ungrouped variables", "name": "bp", "definition": "4", "description": ""}, "ap": {"templateType": "anything", "group": "Ungrouped variables", "name": "ap", "definition": "5", "description": ""}, "cp": {"templateType": "anything", "group": "Ungrouped variables", "name": "cp", "definition": "3", "description": ""}, "dc": {"templateType": "anything", "group": "Ungrouped variables", "name": "dc", "definition": "repeat(random(-15..15),4)", "description": ""}, "fc": {"templateType": "anything", "group": "Ungrouped variables", "name": "fc", "definition": "repeat(random(-50..50),4)", "description": ""}}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>A basic introduction to differentiation</p>"}, "variable_groups": [], "variablesTest": {"condition": "", "maxRuns": 100}, "preamble": {"js": "", "css": ""}, "name": "Differentiation of Basic Polynomial Expressions (with second derivatives)", "functions": {}, "type": "question", "contributors": [{"name": "Nigel Atkins", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/275/"}]}]}], "contributors": [{"name": "Nigel Atkins", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/275/"}]}